1. Cox, Ross and Rubinstein
Binomial Trees
Acedo Fabia Reyes Sorbito Vidamo
2. Report Outline
1
• Overview
2
• General Assumptions
3
• Steps and Formulas
4
• Example
5
• Summary
3. Overview
• A type of binomial asset pricing model first proposed by John
C. Cox, Stephen A. Ross and Mark Rubinstein (1979).
• “Simple and efficient numerical procedure for valuing
options for which premature exercise may be optional”
• “All corporate securities can be interpreted as portfolios of
puts and calls on the asset of the firm.”
• Uses discrete time model of varying price over time of the
underlying financial instrument
• Uses binomial tree of possible price of the underlying asset ;
each nodes valuation is performed iteratively
4. Assumptions
uS with probability p
S
dS with probability q = p ‒ 1
• Underlying asset price S follows a multiplicative binomial
process over discrete period.
• Rate of return on the stock over each period can have two
possible values.
• u and d parameters are constant over the whole tree.
5. Assumptions
• u and d are chosen so that u = 1/d .
• Interest rates are assumed constant, d < Rf < u. It means that
there is no arbitrage opportunity.
• No taxes, transaction cost, or margin requirements
• The underlying doesn't pay dividends over the life of the
option.
6. Steps and Formulas
Step 1. Compute for the Risk free Return
r is the one period rate of return
r = EXP(i*(t/n)) t is term in years
p = (r-d)/(u-d) n is the number of periods
q=1-p p is the risk-neutral probability up move
q is the risk-neutral probability down move
Step 2. Generate the price of the tree
uxS S is the price of underlying asset,
S u is the up move factor with probability p,
dxS d is the down move factor with probability q
7. Steps and Formulas
Step 3. Calculation of option value at each final node
(Backward Induction)
Sn is the computed
At Final Node n:
underlying asset price
If it is a Call Option, then use MAX(0,Sn-K)
at node n
If it is a Put Option, then use MAX(K-Sn,0)
K is the strike price
Step 4. Sequential calculation of the option value at each
preceding node
Cu is the older upper
At other Nodes 0 to n-1 option price
other nodes = [p * Cu + q * Cd] / r Cd is the older lower
option price
8. Example:
Step 1. Compute for the Risk free Return
Stock price [S] $ 60.00 Given
Interest rate [i] 5.00% Given
Strike price [K] $55.00 Given
Term in years [t] 1 Given
Number of periods - quarterly [n] 4 Given
Up move factor [u] 1.05 Given
Down move factor [d] 0.9524 d = 1/u
One period rate of return [r] 1.0126 r = EXP(i*(t/n))
Risk-neutral probability - up move [p] 61.67% p = (r-d)/(u-d)
Risk-neutral probability - down move [q] 38.33% q=1-p
Notes: The price of LDI stock is $60/share and the one-year interest rate is
0.05. We wish to price one-year call option with a strike price of $55. Using a
four-step tree (quarterly) with assumed stock price factor increase of 1.05, we
will compute for the price of the underlying asset and the call option.
9. Example:
Step 2. Generate the price of the tree
Formula: CRR Tree:
0 1 … n 0 1 2 3 4
Suuuu 72.93
Suuu 69.46
Suu Suuud 66.15 66.15
Su Suud 63.00 63.00
S Sud Suudd 60.00 60.00 60.00
Sd Sudd 57.14 57.14
Sdd Suddd 54.42 54.42
Sddd 51.83
Sdddd 49.36
S is the price of underlying asset, S = $ 60
u is the up move factor u = 1.05
d is the down move factor d = 0.9524
n is the number of periods n=4
10. Example:
Step 3. Calculation of option value at each final node
CRR Tree: Binomial Tree for Pricing a $55 Call Option
0 1 2 3 4 0 1 2 3 4
72.93 17.93
69.46
66.15 66.15 11.15
63.00 63.00
60.00 60.00 60.00 5.00
57.14 57.14
54.42 54.42 -
51.83
49.36 -
Given: K = $ 55
At Final Node n:
Sample Computation:
If it is a Call Option, then use MAX(0,Sn-K)
MAX(0, 72.93-55) = 17.93
If it is a Put Option, then use MAX(K-Sn,0)
MAX(0, 66.15-55) = 11.15
11. Example:
Step 4. Calculation of the option value at each preceding node
Binomial Tree for Pricing a $55 Call Option
At other Nodes 0 to n-1
other nodes = [p * Cu + q * Cd] / r 0 1 2 3 4
where
Cu is the older upper option price 17.93
Cd is the older lower option price 15.14
12.51 11.15
Given: p = 1.05, q = 0.9524, r = 1.0126 10.06 8.68
7.87 6.44 5.00
Sample Computation: 4.62 3.04
1.85 -
O31 = [1.05*17.93+0.9524*11.15]/1.0126 -
= 15.14 -
O32 = [1.05*11.15+0.9524*5.00]/1.0126
= 8.68
O21 = [1.05*15.14+0.9524*8.68]/1.0126
= 12.51
12. Summary and Conclusions
• Cox-Ross-Rubinstein Model is one of many available
binomial options pricing models. It is a simplified alternative
numerical method that can be used for practical
computations of complex option values. It assumes a
constant interest rate (risk free return), absence of arbitrage
opportunities and constant probability of underlying assets
upward (u) and downward (d) movement.
• Options priced derived from Cox-Ross-Rubinstein binomial
tree can be used in formulating strategy that will
generate/ lock in pure arbitrage profits if the market price of
an option differs from the value given by the model.
13. References:
• Cox, J.C., Ross S.A, Rubinstein, M., Option Pricing : A Simplified
Approach. (1979). Published in Journal of Finance and
Economics
• Watsham, Terry J., and Parramore, Keith. Quantitative
Methods in Finance. (1997)
• http://investexcel.net/736/binomial-option-pricing-excel/
• http://www.sitmo.com/article/binomial-and-trinomial-trees/
• http://en.wikipedia.org/wiki/Binomial_options_pricing_mode
l
• http://sfb649.wiwi.hu-
berlin.de/fedc_homepage/xplore/tutorials/xlghtmlnode63.ht
ml#bin-fig2
• http://www.terry.uga.edu/~mayhew/Old/chapter9.pdf
Step 1. Binomial model acts similarly to the asset that exists in a risk neutral world.pu+qd = exp(i*∆t) = r, where ∆t = t/nt = term of the optionn= number of periodsIts variance: pu^2 + qd^2 – (exp(i*∆t))^2 =𝜎^2∆tStep 1. Binomial model acts similarly to the asset that exists in a risk neutral world.pu+qd = exp(i*∆t) = r, where ∆t = t/nt = term of the optionn= number of periodsIts variance: pu^2 + qd^2 – (exp(i*∆t))^2 =𝜎^2∆t
Notice that the lattice is symmetrical, that is due to the assumption that d=1/u (ud=1).
